23 11
Article 17.3.4
Journal of Integer Sequences, Vol. 20 (2017),
2 3 6 1
47
Jacobi-Type Continued Fractions for the Ordinary Generating Functions of
Generalized Factorial Functions
Maxie D. Schmidt University of Washington Department of Mathematics
Padelford Hall Seattle, WA 98195
USA
[email protected]
Abstract
The article studies a class of generalized factorial functions and symbolic product se- quences through Jacobi-type continued fractions (J-fractions) that formally enumerate the typically divergent ordinary generating functions of these sequences. The rational convergents of these generalized J-fractions provide formal power series approximations to the ordinary generating functions that enumerate many specific classes of factorial- related integer product sequences. The article also provides applications to a number of specific factorial sum and product identities, new integer congruence relations sat- isfied by generalized factorial-related product sequences, the Stirling numbers of the first kind, and the r-order harmonic numbers, as well as new generating functions for the sequences of binomials,mp−1, among several other notable motivating examples given as applications of the new results proved in the article.
1 Notation and other conventions in the article
1.1 Notation and special sequences
Most of the conventions in the article are consistent with the notation employed within the Concrete Mathematics reference, and the conventions defined in the introduction to the first
article [20]. These conventions include the following particular notational variants:
◮ Extraction of formal power series coefficients. The special notation for formal power series coefficient extraction, [zn] P
kfkzk
:7→fn;
◮ Iverson’s convention. The more compact usage of Iverson’s convention, [i=j]δ ≡ δi,j, in place of Kronecker’s delta function where [n =k= 0]δ≡δn,0δk,0;
◮ Bracket notation for the Stirling and Eulerian number triangles. We use the alternate bracket notation for the Stirling number triangles, n
k
= (−1)n−ks(n, k) and n
k =S(n, k), as well as n
m
to denote the first-order Eulerian number triangle, and n
m
to denote the second-order Eulerian numbers;
◮ Harmonic number sequences. Use of the notation for the first-order harmonic numbers,Hn or Hn(1), which defines the sequence
Hn:= 1 + 1 2+ 1
3+· · ·+ 1 n,
and the notation for the partial sums for the more general cases of ther-order harmonic numbers,Hn(r), defined as
Hn(r):= 1 + 2−r+ 3−r+· · ·+n−r, when r, n≥1 are integer-valued and where Hn(r) ≡0 for all n≤0;
◮ Rising and falling factorial functions. We use the convention of denoting the falling factorial function by xn = x!/(x− n)!, the rising factorial function as xn = Γ(x+ n)/Γ(x), or equivalently by the Pochhammer symbol, (x)n = x(x+ 1)(x + 2)· · ·(x+n−1);
◮ Shorthand notation in integer congruences and modular arithmetic. Within the article the notation g1(n) ≡ g2(n) (mod N1, N2, . . . , Nk) is understood to mean that the congruence, g1(n)≡g2(n) (mod Nj), holds modulo any of the bases, Nj, for 1≤j ≤k.
Within the article, the standard set notation for Z, Q, and R denote the sets of integers, rational numbers, and real numbers, respectively, where the set of natural numbers, N, is defined by N := {0,1,2, . . .} = Z+S
{0}. Other more standard notation for the special functions cited within the article is consistent with the definitions employed within theNIST Handbook reference.
1.2 Mathematica summary notebook document and computational reference information
The article is prepared with a more extensive set of computational data and software routines released as open source software to accompany the examples and numerous other applica- tions suggested as topics for future research and investigation within the article. It is highly encouraged, and expected, that the interested reader obtain a copy of the summary notebook reference and computational documentation prepared in this format to assist with compu- tations in a multitude of special case examples cited as particular applications of the new results.
The prepared summary notebook file, multifact-cfracs-summary.nb , attached to the submission of this manuscript contains the working Mathematica code to verify the formulas, propositions, and other identities cited within the article [21]. Given the length of the article, theMathematica summary notebook included with this submission is intended to help the reader with verifying and modifying the examples presented as applications of the new results cited below. The summary notebook also contains numerical data corresponding to computations of multiple examples and congruences specifically referenced in several places by the applications and tables given in the next sections of the article.
2 Introduction
The primary focus of the new results established by this article is to enumerate new properties of the generalized symbolic product sequences, pn(α, R) defined by (1), which are generated by the convergents to Jacobi-type continued fractions (J-fractions) that represent formal power series expansions of the otherwise divergent ordinary generating functions (OGFs) for these sequences.
pn(α, R) := Y
0≤j<n
(R+αj) [n ≥1]δ+ [n= 0]δ (1)
=R(R+α)(R+ 2α)× · · · ×(R+ (n−1)α) [n ≥1]δ+ [n= 0]δ.
The related integer-valued cases of the multiple factorial sequences, orα-factorial functions, n!(α), of interest in the applications of this article are defined recursively for any fixedα∈Z+ and n∈N by the following equation [20, §2]:
n!(α)=
n·(n−α)!(α), if n >0;
1, if −α < n≤0;
0, otherwise.
(2)
The particular new results studied within the article generalize the known series proved in the references [8,9], including expansions of the series for generating functions enumerating the rising and falling factorial functions, xn = (−1)n(−x)n and xn =x!/(x−n)! =pn(−1, x),
and the Pochhammer symbol, (x)n =pn(1, x), expanded by theStirling numbers of the first kind, n
k
, as A130534:
(x)n=x(x+ 1)(x+ 2)· · ·(x+n−1) [n ≥1]δ+ [n= 0]δ
= Xn
k=1
n k
xk
!
×[n ≥1]δ+ [n= 0]δ.
The generalized rising and falling factorial functions denote the products, (x|α)n = (x)n,α and (x|α)n = (x)n,−α, defined in the reference, where the products, (x)n = (x|1)n and xn = (x|1)n, correspond to the particular special cases of these functions cited above [20,
§2]. Roman’s Umbral Calculus book employs the alternate, and less standard, notation of
x a
n:= xa xa −1
× · · · × xa −n+ 1
to denote the sequences oflower factorial polynomials, and x(n) in place of the Pochhammer symbol to denote the rising factorial polynomials [19,
§1.2]. We do not use this convention within the article.
The generalized product sequences in (1) also correspond to the definition of thePochham- mer α-symbol, (x)n,α =pn(α, x), defined as in ([6], [14, Examples] and [12, cf.§5.4]) for any fixed α 6= 0 and non-zero indeterminate, x ∈ C, by the following analogous expansions in- volving the generalized α-factorial coefficient triangles defined in (4) of the next section of this article [20, §3]:
(x)n,α =x(x+α)(x+ 2α)· · ·(x+ (n−1)α) [n≥1]δ+ [n= 0]δ
= Xn k=1
n k
αn−kxk
!
×[n ≥1]δ+ [n= 0]δ
= Xn k=0
n+ 1 k+ 1
α
(x−1)k
!
×[n ≥1]δ+ [n= 0]δ.
We are especially interested in using the new results established in this article to formally generate the factorial-function-like product sequences,pn(α, R) andpn(α, βn+γ), for some fixed parameters α, β, γ ∈ Q, when the initially fixed symbolic indeterminate, R, depends linearly on n. The particular forms of the generalized product sequences of interest in the applications of this article are related to theGould polynomials,Gn(x;a, b) = x−xan· x−ban
n
, in the form of the following equation ([20, §3.4.2],[19, §4.1.4]):
pn(α, βn+γ) = (−α)n+1
γ−α−β ×Gn+1(γ−α−β;−β,−α). (3) Whereas the first results proved in the articles [8,9] are focused on establishing properties of divergent forms of the ordinary generating functions for a number of special sequence cases through more combinatorial interpretations of these continued fraction series, the emphasis in this article is more enumerative in flavor. The new identities involving the integer-valued cases of the multiple,α-factorial functions,n!(α), defined in (2) obtained by this article extend
the study of these sequences motivated by the distinct symbolic polynomial expansions of these functions originally considered in the reference [20]. This article extends a number of the examples considered as applications of the results from the 2010 article [20] briefly summarized in the next subsection.
2.1 Polynomial expansions of generalized α-factorial functions
For any fixed integerα≥1 andn, k ∈N, the coefficients defined by the triangular recurrence relation in (4) provide one approach to enumerating the symbolic polynomial expansions of the generalized factorial function product sequences defined as special cases of (1) and (2).
n k
α
= (αn+ 1−2α) n−1
k
α
+
n−1 k−1
α
+ [n=k = 0]δ (4)
The combinatorial interpretations of these coefficients as generalized Stirling numbers of the first kind motivated in the reference [20] leads to polynomial expansions in n of the multiple factorial function sequence variants in (2) that generalize the known formulas for the single and double factorial functions, n! and n!!, involving the unsigned Stirling numbers of the first kind, n
k
= n
k
1 = (−1)n−ks(n, k), which are expanded in the forms of the following equations ([12,§6], [17, §26.8], A000142,A006882):
n! = Xn m=0
n m
(−1)n−mnm, ∀n ≥1 n!! =
Xn m=0
⌊n+12 ⌋ m
(−2)⌊n+12 ⌋−mnm, ∀n ≥1 n!(α) =
Xn m=0
⌈n/α⌉ m
(−α)⌈nα⌉−mnm, ∀n ≥1, α ∈Z+. (5) The Stirling numbers of the first kind similarly provide non-polynomial exact finite sum formulas for the single and double factorial functions in the following forms forn ≥1 where (2n)!! = 2n×n! ([12,§6.1],[4, §5.3]):
n! = Pn
k=0
n
k
and (2n−1)!! = Pn
k=0
n
k
2n−k.
Related finite sums generating the single factorial, double factorial, andα-factorial functions are expanded respectively through thefirst-order Euler numbers, n
m
, asn! =Pn−1 k=0
n
k
[17,
§26.14(iii)], through thesecond-order Euler numbers,n
m
, as (2n−1)!! =Pn−1 k=0
n
k
[12,
§6.2], and by the generalized cases of these triangles [20, §6.2.4].
The polynomial expansions of the first two classical sequences of the previous equations in (5) are generalized to the more general α-factorial function cases through the triangles
defined as in (4) from the reference [20] through the next explicit finite sum formulas when n≥1.
n!(α) = Xn m=0
⌊n−1+αα ⌋+ 1 m+ 1
α
(−1)⌊n−α1+α⌋−m(n+ 1)m, ∀n ≥1, α∈Z+ (6) The polynomial expansions innof the generalizedα-factorial functions, (αn−d)!(α), for fixed α ∈ Z+ and integers 0 ≤ d < α, are obtained similarly from (6) through the generalized coefficients in (4) as follows [20, cf. §2]:
(αn−d)!(α) = (α−d)× Xn m=1
n m
α
(−1)n−m(αn+ 1−d)m−1 (7)
=
Xn m=0
n+ 1 m+ 1
α
(−1)n−m(αn+ 1−d)m, ∀n ≥1, α ∈Z+,0≤d < α.
A binomial-coefficient-themed phrasing of the products underlying the expansions of the more general factorial function sequences of this type (each formed by dividing through by a normalizing factor of n!) is suggested in the following expansions of these coefficients by the Pochhammer symbol [12,§5]:
s−1
α
n
= (−1)n n! ·
s−1 α
n
= 1
αn·n!
n−1
Y
j=0
(s−1−αj). (8)
When the initially fixed indeterminate s :=sn is considered modulo α in the form of sn :=
αn+d for some fixed least integer residue, 0 ≤d < α, the prescribed setting of this offset d completely determines the numerical α-factorial function sequences of the forms in (7) generated by these products (see, for example, the examples cited below in Section 3.2 and the tables given in the reference [20, cf.§6.1.2, Table 6.1]).
For any lower index n≥ 1, the binomial coefficient formulation of the multiple factorial function products in (8) provides the next several expansions by the exponential generating functions for the generalized coefficient triangles in (4), and their corresponding generalized Stirling polynomial analogs,σk(α)(x), defined in the references ([20, §5],[12, cf.§6, §7.4]):
s−1
α
n
= Xn m=0
n+ 1 n+ 1−m
α
(−1)msn−m
αnn! (9)
= Xn m=0
(−1)m·(n+ 1)σm(α)(n+ 1)
αm × (s/α)n−m (n−m)!
s−1 α
n
= [zn] e(s−1+α)z/α
−ze−z e−z −1
n+1!
(10)
= [znwn]
−z·e(s−1+α)z/α 1 +wz−ez
.
The generalized forms of the Stirling convolution polynomials, σn(x) ≡ σn(1)(x), and the α- factorial polynomials, σ(α)n (x), studied in the reference are defined for each n ≥ 0 through the triangle in (4) as follows [20, §5.2]:
x·σn(α)(x) :=
x x−n
α
(x−n−1)!
(x−1)! Generalized Stirling Polynomials
= [zn]
e(1−α)z
αzeαz eαz−1
x .
A more extensive treatment of the properties and generating function relations satisfied by the triangular coefficients defined by (4), including their similarities to the Stirling number triangles, Stirling convolution polynomial sequences, and the generalized Bernoulli polynomi- als, among relations to several other notable special sequences, is provided in the references.
2.2 Divergent ordinary generating functions approximated by the convergents to infinite Jacobi-type and Stieltjes-type contin- ued fraction expansions
2.2.1 Infinite J-fraction expansions generating the rising factorial function Another approach to enumerating the symbolic expansions of the generalized α-factorial function sequences outlined above is constructed as a new generalization of the continued fraction series representations of the ordinary generating function for the rising factorial function, or Pochhammer symbol, (x)n= Γ(x+n)/Γ(x), first proved by Flajolet [8, 9]. For any fixed non-zero indeterminate, x ∈ C, the ordinary power series enumerating the rising factorial sequence is defined through the next infinite Jacobi-type J-fraction expansion [8,
§2, p. 148]:
R0(x, z) := X
n≥0
(x)nzn = 1
1−xz− 1·xz2
1−(x+ 2)z− 2(x+ 1)z2
1−(x+ 4)z− 3(x+ 2)z2
· · · .
(11)
Since we know symbolic polynomial expansions of the functions, (x)n, through the Stirling numbers of the first kind, we notice that the terms in a convergent power series defined by (11) correspond to the normalized coefficients of the following well-known two-variable “double”, or “super”, exponential generating functions (EGFs) for the Stirling number triangle whenx is taken to be a fixed, formal parameter with respect to these series ([12, §7.4],[17,§26.8(ii)], [8, cf. Prop. 9]):
X
n≥0
(x)nzn
n! = 1
(1−z)x and X
n≥0
xn· zn
n! = (1 +z)x.
For natural numbers m ≥ 1 and fixed α ∈ Z+, the coefficients defined by the generalized triangles in (4) are enumerated similarly by the generating functions [20, cf.§3.3]
X
m,n≥0
n+ 1 m+ 1
α
xmzn
n! = (1−αz)−(x+1)/α. (12)
When x depends linearly on n, the ordinary generating functions for the numerical facto- rial functions formed by (x)n do not converge for z 6= 0. However, the convergents of the continued fraction representations of these series still lead to partial, truncated series approx- imations generating these generalized product sequences, which in turn immediately satisfy a number of combinatorial properties, recurrence relations, and other established integer congruence properties implied by the rational convergents to the first continued fraction expansion given in (11).
2.2.2 Examples
Two particular divergent ordinary generating functions for the single factorial function se- quences, f1(n) := n! and f2(n) := (n+ 1)!, are cited in the references as examples of the Jacobi-type J-fraction results proved in Flajolet’s articles ([8,9],[16, cf.§5.5]). The next pair of series expansions serve to illustrate the utility to enumerating each sequence formally with respect toz required by the results in this article [8, Thm. 3A; Thm. 3B].
F1,∞(z) := X
n≥0
n!·zn = 1
1−z− 12·z2 1−3z−22z2
· · ·
Single Factorial J-Fractions
F2,∞(z) := X
n≥0
(n+ 1)!·zn = 1
1−2z− 1·2z2 1−4z−2·3z2
· · ·
In each of these respective formal power series expansions, we immediately see that for each finite h ≥ 1, the hth convergent functions, denoted Fi,h(z) for i = 1,2, satisfy fi(n) = [zn]Fi,h(z) whenever 1 ≤ n < 2h. We also have that fi(n) ≡ [zn]Fi,h(z) (mod p) for any n≥0 whenever p is a divisor ofh ([9],[16, cf.§5]).
Similar expansions of other factorial-related continued fraction series are given in the references ([8],[16, cf.§5.9]). For example, the next known Stieltjes-type continued fractions (S-fractions), formally generating the double factorial function, (2n−1)!!, and the Catalan numbers, Cn, respectively, are expanded through the convergents of the following infinite
continued fractions ([8, Prop. 5; Thm. 2], [16, §5.5], A001147,A000108):
F3,∞(z) :=X
n≥0
1·3· · ·(2n−1)
| {z } (2n−1)!!
×z2n = 1 1− 1·z2
1− 2·z2 1− 3·z2
· · ·
Double Factorial S-Fractions
F4,∞(z) :=X
n≥0
2n(2n−1)!!
(n+ 1)!
| {z } Cn= 2nn 1
(n+1)
×z2n = 1 1− z2
1− z2 1− z2
· · ·.
For comparison, some related forms of regularized ordinary power series inz generating the single and double factorial function sequences from the previous examples are stated in terms of the incomplete gamma function, Γ(a, z) =R∞
z ta−1e−tdt, as follows [17, §8]:
X
n≥0
n!·zn =−e−1/z z ×Γ
0,−1
z
X
n≥0
(n+ 1)!·zn =−e−1/z z2 ×Γ
−1,−1 z
X
n≥1
(2n−1)!!·zn =− e−1/2z (−2z)3/2 ×Γ
−1 2,− 1
2z
. (13)
Sincepn(α, R) = αn(R/α)n, the exponential generating function for the generalized product sequences corresponds to the series ([12, cf.§7.4, eq. (7.55)],[14]):
Pb(α, R;z) :=
X∞ n=0
pn(α, R)zn
n! = (1−αz)−R/α, Generalized Product Sequence EGFs where for each fixed α ∈ Z+ and 0 ≤ r < α, we have the identities, (αn−r)!(α) = pn(α, α−r) = αn 1− αr
n. The form of this exponential generating function then leads to the next forms of the regularized sums by applying aLaplace transform to the generating functions in the previous equation (see Remark 19) ([7, §B.14],[17, cf.§8.6(i)]).
Beα,−r(z) :=X
n≥0
(αn−r)!(α)zn
= Z ∞
0
e−t
(1−αtz)1−r/αdt
= e−αz1
(−αz)1−r/α ×Γ r
α,− 1 αz
The remarks given in Section4.3 suggest similar approximations to theα-factorial functions generated by the generalized convergent functions defined in the next section, and their relations to the confluent hypergeometric functions and the associated Laguerre polynomial sequences ([17, cf.§18.5(ii)], [19, §4.3.1]).
2.3 Generalized convergent functions generating factorial-related integer product sequences
2.3.1 Definitions of the generalized J-fraction expansions and the generalized convergent function series
We state the next definition as a generalization of the result for the rising factorial function due to Flajolet cited in (11) to form the analogous series enumerating the multiple, or α- factorial, product sequence cases defined by (1) and (2).
Definition 1(Generalized J-Fraction Convergent Functions). Suppose that the parameters h∈N,α∈Z+andR:=R(n) are defined in the notation of the product-wise sequences from (1). For h ≥ 0 and z ∈ C, we let the component numerator and denominator convergent functions, denoted FPh(α, R;z) and FQh(α, R;z), respectively, be defined by the recurrence relations in the next equations.
FPh(α, R;z) :=
(1−(R+2α(h−1))z) FPh−1(α,R;z)−α(R+α(h−2))(h−1)z2FPh−2(α,R;z) if h≥2;
1, if h= 1;
0, otherwise.
(14)
FQh(α, R;z) :=
(1−(R+2α(h−1))z) FQh
−1(α,R;z)−α(R+α(h−2))(h−1)z2FQh
−2(α,R;z) if h≥2;
1−Rz, if h= 1;
1, if h= 0;
0, otherwise.
(15) The corresponding convergent functions, Convh(α, R;z), defined in the next equation then provide the rational, formal power series approximations in z to the divergent ordinary gen- erating functions of many factorial-related sequences formed as special cases of the symbolic
products in (1).
Convh(α, R;z) := 1
1−R·z− αR·z2
1−(R+ 2α)·z− 2α(R+α)·z2
1−(R+ 4α)·z− 3α(R+ 2α)·z2
· · ·
1−(R+ 2(h−1)α)·z
= FPh(α, R;z) FQh(α, R;z) =
2hX−1 n=0
pn(α, R)zn+ X∞ n=2h
e
eh,n(α, R)zn (16)
The first series coefficients on the right-hand-side of (16) generate the products, pn(α, R), from (1), where the remaining forms of the power series coefficients,eeh,n(α, R), correspond to
“error terms” in the truncated formal series approximations to the exact sequence generating functions obtained from these convergent functions, which are defined such thatpn(α, R)≡ e
eh,n(α, R) (mod h) for all h≥2 and n≥2h.
2.3.2 Properties of the generalized J-fraction convergent functions
A number of the immediate, noteworthy properties satisfied by these generalized convergent functions are apparent from inspection of the first few special cases provided in Table 9.1 (page 71) and in Table 9.2 (page 72). The most important of these properties relevant to the new interpretations of the α-factorial function sequences proved in the next sections of the article are briefly summarized in the points stated below.
◮ Rationality of the convergent functions in α, R, and z:
For any fixed h ≥ 1, it is easy to show that the component convergent functions, FPh(z) and FQh(z), defined by (14) and (15), respectively, are polynomials of finite degree in each ofz, R, andα satisfying
degz,R,α
FPh(α, R;z) =h−1 and degz,R,α
FQh(α, R;z) =h.
For any h, n ∈ Z+, if R := R(n) denotes some linear function of n, the product sequences, pn(α, R), generated by the generalized convergent functions always corre- spond to polynomials in n (in R) of predictably finite degree with integer coefficients determined by the choice ofn ≥1.
◮ Expansions of the denominator convergent functions by special functions:
For all h ≥ 0, and fixed non-zero parameters α and R, the power series in z gen- erated by the generalized hth convergents, Convh(α, R;z), are characterized by the representations of the convergent denominator functions, FQh(α, R;z), through the
confluent hypergeometric functions,U(a, b, w) and M(a, b, w), and the associated La- guerre polynomial sequences, L(β)n (x), as follows (see Section 4.3) ([17, §13; §18], [19,
§4.3.1]):
zh·FQh α, R;z−1
| {z } FQfh(α, R;z)
=αh×U
−h,R α,z
α
(17)
= (−α)h(R/α)h×M
−h,R α, z
α
= (−α)h·h!×L(R/αh −1)z α
.
The special function expansions of the reflected convergent denominator function se- quences above lead to the statements of addition theorems, multiplication theorems, and several additional auxiliary recurrence relations for these functions proved in Sec- tion 5.1.
◮ Corollaries: New exact formulas and congruence properties for theα-factorial functions and the generalized product sequences:
If some ordering of the h zeros of (17) is fixed at each h ≥ 1, we can define the next sequences which form special cases of the zeros studied in the references [10, 3]. In particular, each of the following special zero sequence definitions given as ordered sets, or ordered lists of zeros, provide factorizations over z of the denominator sequences, FQh(α, R;z), parameterized by α and R:
(ℓh,j(α, R))hj=1 :=n
zj :αh ×U
−h, R/α, z α
= 0, 1≤j ≤ho
Special Function Zeros
=n
zj :αh ×L(R/αh −1)z α
= 0, 1≤j ≤ho .
Let the sequences, ch,j(α, R), denote a shorthand for the coefficients corresponding to an expansion of the generalized convergent functions, Convh(α, R;z), by partial fractions in z, i.e., the coefficients defined so that [17, §1.2(iii)]
Convh(α, R;z) :=
Xh j=1
ch,j(α, R) (1−ℓh,j(α, R)·z).
Forn≥1 and any fixed integerα 6= 0, these rational convergent functions provide the following formulas exactly generating the respective sequence cases in (1) and (2):
pn(α, R) = Xn
j=1
cn,j(α, R)×ℓn,j(α, R)n (18) n!(α) =
Xn j=1
cn,j(−α, n)×ℓn,j(−α, n)⌊n−α1⌋.
The corresponding congruences satisfied by each of these generalized sequence cases obtained from thehth convergent function expansions in z are stated similarly modulo any prescribed integersh≥2 and fixed α≥1 in the next forms.
pn(α, R)≡ Xh
j=1
ch,j(α, R)×ℓh,j(α, R)n (mod h) (19)
n!(α) ≡ Xh
j=1
ch,j(−α, n)×ℓh,j(−α, n)⌊n−1α ⌋ (mod h, hα,· · · , hαh) Section3.3and Section 6.2 provide several particular special case examples of the new congruence properties expanded by (19).
3 New results proved within the article
3.1 A summary of the new results and outline of the article topics
J-Fractions for generalized factorial function sequences (Section 4 on page 20) The article contains a number of new results and new examples of applications of the results from Section 4 in the next subsections. The Jacobi-type continued fraction expansions formally enumerating the generalized factorial functions, pn(α, R), proved in Section 4 are new, and moreover, follow easily from the known continued fractions for the series generating the rising factorial function, (x)n =pn(1, x), established by Flajolet [8].
Properties of the generalized convergent functions (Section 5 on page 24)
In Section 5 we give proofs of new properties, expansions, recurrence relations, and exact closed-form representations by special functions satisfied by the convergent numerator and denominator subsequences, FPh(α, R;z) and FQh(α, R;z). The consideration of the conver- gent function approximations to these infinite continued fraction expansions is a new topic not previously explored in the references which leads to new integer congruence results for factorial functions as well as new applications to generating function identities enumerating factorial-related integer sequences.
Applications and motivating examples (Section 6 on page 32)
Specific consequences of the convergent function properties we prove in Section 5 of the article include new congruences for and new rational generating function representations enumerating the Stirling numbers of the first kind,n
k
, modulo fixed integers m≥2 and the scaledr-order harmonic numbers,n!r·Hn(r), modulom, which are easily extended to formulate analogous congruence results for theα-factorial functions,n!(α), and the generalized factorial
product functions, pn(α, R). These particular special cases of the new congruence results are proved in Section 6.2. Section 6.2 also contains proofs of new representations of exact formulas for factorial functions expanded by the special zeros of the Laguerre polynomials and confluent hypergeometric functions.
The subsections of Section 6 also provide specific sequence examples and new sequence generating function identities that demonstrate the utility and breadth of new applications implied by the convergent-based rational and hybrid-rational generating functions we rig- orously treat first in Section 4 and Section 5. For example, in Section 6.3, we are the first to notice several specific integer sequence applications of new convergent-function- based Hadamard product identities that effectively provide truncated series approximations to the formal Laplace-Borel transformation where multiples of the rational convergents, Convn(α, R;z), generate the sequence multiplier,n!, in place of more standard integral rep- resentations of the transformation. In Section6.4through Section6.7, we focus on expanding particular examples of convergent-function-based generating functions enumerating special factorial-related integer sequences and combinatorial identities. The next few subsections of the article provide several special case examples of the new applications, new congruences, and other examples of the new results established in Section 6.
3.2 Examples of factorial-related finite product sequences enu- merated by the generalized convergent functions
3.2.1 Generating functions for arithmetic progressions of the α-factorial func- tions
There are a couple of noteworthy subtleties that arise in defining the specific numerical forms of the α-factorial function sequences defined by (2) and (6). First, since the generalized convergent functions generate the distinct symbolic products that characterize the forms of these expansions, we see that the following convergent-based enumerations of the multiple factorial sequence variants hold at each α, n ∈ Z+, and some fixed choice of the prescribed offset, 0≤d < α:
(αn−d)!(α) = (−α)n· d
α −n
| {z n} pn(−α, αn−d)
= [zn] Convn(−α, αn−d;z) (20)
=αn·
1− d α
| {z n} pn(α, α−d)
= [zn] Convn(α, α−d;z).
For example, some variants of the arithmetic progression sequences formed by the single factorial and double factorial functions, n! and n!!, in Section 6.4 are generated by the particular shifted inputs to these functions highlighted by the special cases in the next
equationsA000142, A000165, A001147:
(n!)∞n=1 = ((1)n)∞n=1 −−−−−→A000142 (1,2,6,24,120,720,5040, . . .) ((2n)!!)∞n=1 = (2n·(1)n)∞n=1 −−−−−→A001147 (2,8,48,384,3840,46080, . . .) ((2n−1)!!)∞n=1 = (2n·(1/2)n)∞n=1 −−−−−→A000165 (1,3,15,105,945,10395, . . .). The next few special case variants of the α-factorial function sequences corresponding to α:= 3,4, also expanded in Section 6.4, are given in the following sequence forms (A032031, A008544,A007559, A047053, A007696):
((3n)!!!)∞n=1 = (3n·(1)n)∞n=1 −−−−−→A032031 (3,18,162,1944,29160, . . .) ((3n−1)!!!)∞n=1 = (3n·(2/3)n)∞n=1 −−−−−→A008544 (2,10,80,880,12320,209440, . . .) ((3n−2)!!!)∞n=1 = (3n·(1/3)n)∞n=1 −−−−−→A007559 (1,4,28,280,3640,58240, . . .)
(4n)!(4)∞
n=0 = (4n·(1)n)∞n=0 −−−−−→A047053 (1,4,32,384,6144,122880, . . .) (4n+ 1)!(4)
∞
n=0 = (4n·(5/4)n)∞n=0 −−−−−→A007696 (1,5,45,585,9945,208845, . . .). For each n ∈ N and prescribed constants r, c ∈ Z defined such that c | n+r, we also ob- tain rational convergent-based generating functions enumerating the modified single factorial function sequences given by
n+r c
!
= [zn] Convh−c, n+r;z c
+hr c = 0i
δ[n = 0]δ, ∀ h≥ ⌊(n+r)/c⌋. (21) 3.2.2 Generating functions for multi-valued integer product sequences
Likewise, given any n ≥ 1 and fixed α ∈Z+, we can enumerate the somewhat less obvious full forms of the generalizedα-factorial function sequences defined piecewise for the distinct residues, n ∈ {0,1, . . . , α−1}, modulo α by (2) and in (20). The multi-valued products defined by (1) for these functions are generated as follows:
n!(α)=
z⌊(n+α−1)/α⌋
Convn(−α, n;z) (22)
= [zn] X
0≤d<α
z−d·Convn(α, α−d;zα)
!
= [zn+α−1]
1−zα
1−z ×Convn(−α, n;zα)
. (23)
The complete sequences over the multi-valued symbolic products formed by the special cases of the double factorial function, the triple factorial function, n!!!, the quadruple factorial
function,n!!!! = n!(4), thequintuple factorial (5-factorial) function,n!(5), and the 6-factorial function,n!(6), respectively, are generated by the convergent generating function approxima- tions expanded in the next equations (A006882, A007661, A007662, A085157,A085158).
(n!!)∞n=1 =
z⌊(n+1)/2⌋
Convn(−2, n;z)∞ n=1
A006882
−−−−−→ (1,2,3,8,15,48,105,384, . . .) (n!!!)∞n=1 =
z⌊(n+2)/3⌋
Convn(−3, n;z)∞ n=1
A007661
−−−−−→ (1,2,3,4,10,18,28,80,162, . . .) n!(4)
∞
n=1 =
z⌊(n+3)/4⌋
Convn(−4, n;z)∞ n=1
A007662
−−−−−→ (1,2,3,4,5,12,21,32,45, . . .) n!(5)
∞
n=1 =
z⌊(n+4)/5⌋
Convn(−5, n;z)∞ n=1
A085157
−−−−−→ (1,2,3,4,5,6,14,24,36,50, . . .) n!(6)∞
n=1 =
z⌊(n+5)/6⌋
Convn(−6, n;z)∞ n=1
A085158
−−−−−→ (1,2,3,4,5,6,7,16,27,40, . . .) 3.2.3 Examples of new convergent-based generating function identities for bi-
nomial coefficients
The rationality inzof the generalized convergent functions, Convh(α, R;z), for allh≥1 also provides several of the new forms of generating function identities for many factorial-related product sequences and related expansions of the binomial coefficients that are easily proved from the diagonal-coefficient, or Hadamard product, generating function results established in Section 6.3. For example, for natural numbers n ≥ 1, the next variants of the binomial- coefficient-related product sequences are enumerated by the following coefficient identities A009120,A001448:
(4n)!
(2n)! = 44n❩(1)❩❩n❩❩❩
2 4
n 1 4
n 3 4
n
22n❩(1)❩❩n❩❩❩
1 2
n
(24)
= 4n×4n(1/4)n×4n(3/4)n
= [x01zn]
Convn
4,3;4z x1
Convn(4,1;x1)
= 4n×(4n−3)!(4)(4n−1)!(4)
= [x01zn]
Convn
−4,4n−3;4z x1
Convn(−4,4n−1;x1) 4n
2n
= [x01x02zn] Convn
4,3;4z x2
Convn
4,1;x2
x1
× cosh (√ x1)
| {z } Eb2(x1) =E2,1(x1)
!
= [x01x02zn] Convn
−4,4n−3;4z x2
Convn
−4,4n−1;x2 x1
×cosh (√ x1)
| {z } E2,1(x1)
! .
The examples given in Section6.3.2 provide examples of related constructions of the hybrid rational convergent-based generating function products that generate the central binomial coefficients and several other notable cases of related sequence expansions. We can similarly generate the sequences of binomials, (a+b)n and cn−1 for fixed non-zero a, b, c∈R, which we consider in Section 6.7, using the binomial theorem and a rational convergent-based approximation to the formal Laplace-Borel transform as follows:
(a+b)n=n!× Xn k=0
ak
k! · bn−k (n−k)!
= [x0][zn] Convn
1,1;z x
e(a+b)x
= [x0][zn] Convn
−1, n;z x
e(a+b)x
cn−1 = n!×
n−1
X
k=0
(c−1)k+1
(k+ 1)! · 1 (n−1−k)!
= [x0][zn] Convn 1,1;z
x
e(c−1)x−1 ex
= [x0][zn] Convn
−1, n;z x
e(c−1)x−1 ex.
3.3 Examples of new congruences for the α-factorial functions, the Stirling numbers of the first kind, and the r-order harmonic number sequences
3.3.1 Congruences for the α-factorial functions modulo 2
For any fixed α ∈ Z+ and natural numbers n ≥ 1, the generalized multiple, α-factorial functions,n!(α), defined by (2) satisfy the following congruences modulo 2 (and 2α):
n!(α) ≡ n 2
n−α+p
α(α−n)⌊n−1α ⌋ +
n−α−p
α(α−n)⌊n−1α ⌋
(mod 2,2α)
= [zn]
(zα−1)(1 + (2α−n)zα) z(1−z) ((α−n)(n·zα−2)zα−1)
. (25)
Given that the definition of the single factorial function implies that n!≡0 (mod 2) when- ever n≥2, the statement of (25) provides somewhat less obvious results for the generalized α-factorial function sequence cases whenα ≥2. Table9.3(page75) provides specific listings of the result in (25) satisfied by the α-factorial functions, n!(α), for α := 1,2,3,4. The cor- responding, closely-related new forms of congruence properties satisfied by these functions expanded by exact algebraic formulas modulo 3 (3α) and modulo 4 (4α) are also cited as special cases in the examples given in the next subsection (see Section6.2.2).
3.3.2 New forms of congruences for the α-factorial functions modulo 3, modulo 4, and modulo 5
To simplify notation, we first define the next shorthand for the respective (distinct) roots, r(α)p,i(n) for 1 ≤ i ≤ p, corresponding to the special cases of the convergent denominator functions, FQp(α, R;z), factorized over z for any fixed integers n, α ≥ 1 when p := 3,4,5 [17, §1.11(iii); cf. §4.43]:
r3,i(α)(n)3
i=1 :=
zi :z3i −3zi2(2α+n) + 3zi(α+n)(2α+n)
−n(α+n)(2α+n) = 0, 1≤i≤3 r(α)4,j(n)4
j=1 :=
zj :zj4−4zj3(3α+n) + 6zj2(2α+n)(3α+n)−4zj(α+n)(2α+n)(3α+n) +n(α+n)(2α+n)(3α+n) = 0, 1≤j ≤4
r5,k(α)(n)5
k=1 :=
zk:zk5−5(4α+n)zk4+ 10(3α+n)(4α+n)zk3
−10(2α+n)(3α+n)(4α+n)zk2 + 5(α+n)(2α+n)(3α+n)(4α+n)zk
−n(α+n)(2α+n)(3α+n)(4α+n) = 0, 1≤k≤5 . (26) Similarly, we define the following functions for any fixed α ∈ Z+ and n ≥ 1 to simplify the notation in stating next the congruences in (27) below:
Re(α)3 (n) :=
6α2+α
6r(3,1−α)(n)−4n +
n−r3,1(−α)(n)
2
r3,1(−α)(n)⌊n−1α ⌋+1 r(3,1−α)(n)−r(3,2−α)(n) r3,1(−α)(n)−r3,3(−α)(n)
+
6α2+α
6r3,3(−α)(n)−4n +
n−r(3,3−α)(n)
2
r(3,3−α)(n)⌊n−1α ⌋+1 r3,3(−α)(n)−r3,1(−α)(n) r(3,3−α)(n)−r(3,2−α)(n)
+
6α2+α
6r3,2(−α)(n)−4n +
n−r(3,2−α)(n)
2
r(3,2−α)(n)⌊n−α1⌋+1 r3,2(−α)(n)−r3,1(−α)(n) r(3,2−α)(n)−r(3,3−α)(n)
C4,i(α)(n) := 24α3 −18α2
n−2·r4,i(−α)(n) +α
7n−12·r4,i(−α)(n) n−r4,i(−α)(n)
−
n−r4,i(−α)(n)3
, for 1≤i≤4 C5,k(α)(n) := 120α4+ 2α2
23n2−79n·r(5,k−α)(n) + 60·r5,k(−α)(n)2
+ 48α3(2n−5·r5,k(−α)(n)) +α(11n−20r5,k(−α)(n))(n−r(5,k−α)(n))2+ (n−r(5,k−α)(n))4, for 1≤k ≤5.
For fixed α ∈Z+ and n ≥ 0, we obtain the following analogs to the first congruence result modulo 2 expanded by (25) for theα-factorial functions,n(α), whenn ≥1 (see Section6.2.2):
n!(α)≡Re(α)3 (n) (mod 3,3α) (27)
n!(α)≡ X
1≤i≤4
C4,i(α)(n) Q
j6=i
r(4,i−α)(n)−r(4,j−α)(n)r4,i(−α)(n)⌊n+αα−1⌋
| {z } :=R4(α)(n)
(mod 4,4α)
n!(α)≡ X
1≤k≤5
C5,k(α)(n) Q
j6=k
r(5,k−α)(n)−r(5,j−α)(n)r(5,k−α)(n)⌊n+α−1α ⌋
| {z } :=R(α)5 (n)
(mod 5,5α).
Several particular concrete examples illustrating the results cited in (25) modulo 2 (and 2α), and in (27) modulo p(and pα) forp:= 3,4,5, corresponding to the first few cases of α≥1 and n ≥1 appear in Table 9.3 (page 75) Further computations of the congruences given in (27) modulo pαi (for some 0≤i≤p) are contained in theMathematica summary notebook included as a supplementary file with the submission of this article (see Section1.2 and the reference document [21]). The results in Section6.2 provide statements of these new integer congruences for fixed α 6= 0 modulo any integers p≥ 2. The analogous formulations of the new relations for the factorial-related product sequences modulo any p and pα are easily established for the subsequent cases of integersp≥6 from the partial fraction expansions of the convergent functions, Convh(α, R;z), cited in the particular listings in Table 9.1 (page 71) and in Table 9.2 (page 72), and through the generalized rational convergent function properties proved in Section5.
3.3.3 New congruence properties for the Stirling numbers of the first kind The results given in Section 6.2 also provide new congruences for the generalized Stirling number triangles in (4), as well as several new forms of rational generating functions that enumerate the scaled factorial-power variants of ther-order harmonic numbers, (n!)r×Hn(r), modulo integers p ≥ 2 [12, §6.3]. For example, the known congruences for the Stirling numbers of the first kind proved by the generating function techniques enumerated in the reference [25,§4.6] imply the next new congruence results satisfied by the binomial coefficients
modulo 2 (see Section 6.2.1) A087755.
⌊n2⌋ m− ⌈n2⌉
[1≤m ≤6]δ
≡[n > m]δ×
2n
4 , if m = 1;
3·2n
16 (n−1), if m = 2;
2n
128(9n−20)(n−1), if m = 3;
2n
512(3n−10)(3n−7)(n−1), if m = 4;
2n
8192(27n3−279n2+ 934n−1008)(n−1), if m = 5;
2n
163840(9n2−71n+ 120)(3n−14)(3n−11)(n−1), if m = 6;
0, otherwise.
+ [1≤m≤6]δ[n =m]δ (mod 2)
3.3.4 New congruences and rational generating functions for the r-order har- monic numbers
The next results state several additional new congruence properties satisfied by the first- order, second-order, and third-order harmonic number sequences, each expanded by the rational generating functions enumerating these sequences modulo the first few small cases of integer-valued p constructed from the generalized convergent functions in Section 6.2.1 (A001008, A002805, A007406,A007407, A007408,A007409).
(n!)3×Hn(3) ≡[zn]
z(1−7z+49z2−144z3+192z4)
(1−8z)2
(mod 2) (n!)2×Hn(2) ≡[zn]
z(1−61z+1339z2−13106z3+62284z4−144264z5+151776z6−124416z7+41472z8)
(1−6z)3(1−24z+36z2)2
(mod 3) (n!)×Hn(1) ≡[zn+1]
36z2−48z+325
576 + 17040z2+1782z+6467
576(24z3−36z2+12z−1) + 78828z2−33987z+3071 288(24z3−36z2+12z−1)2
(mod 4)
≡[zn ]
3z−4
48 +96(24z1300z3−2+890z+94736z2+12z−1) + 24568z2−10576z+955
96(24z3−36z2+12z−1)2
(mod 4)
≡[zn−1]
1
16+ 48(24z−96z32−+794z+39736z2+12z−1) +24(24z5730z32−2453z+221
−36z2+12z−1)2
(mod 4).
4 The Jacobi-type J-fractions for generalized factorial function sequences
4.1 Enumerative properties of Jacobi-type J-fractions
To simplify the exposition in this article, we adopt the notation for the Jacobi-type continued fractions, or J-fractions, in ([8, 9], [17, cf. §3.10], [16, cf. §5.5]). Given some application- specific choices of the prescribed sequences, {ak, bk, ck}, we consider the formal power series
